HEAVY MOVABLE STRUCTURES MOVABLE BRIDGES AFFILIATE 3RD BIENNIAL SYMPOSIUM ST. PETERSBURG HILTON & TOWERS ST. PETERSBURG, FLORIDA

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1 HEAVY MOVABLE STRUCTURES MOVABLE BRIDGES AFFILIATE 3RD BIENNIAL SYMPOSIUM NOVEMBER 12TH - 7_5TH, 1990 ST. PETERSBURG HILTON & TOWERS ST. PETERSBURG, FLORIDA SESSION WORKSHOP NOTES Session (5-3) "Design Criteria for Shafts and Open Gear'g on Movable Bridges", L.Borden, Modjeski & Masters, Pa. Disclaimer It rs the policy of t e Aff:::a::w :a?;av..de a?.?an far :nfornat:x 1--,er;hange. It 3CZS XGT arcoaaate, recomend+r enaorse any of rne :nf9rnat:on :r:erchar.~ea as I: rs;ares :a ;es:gn prlncrples, processes, or& ro ucts preseqtea a tne ympos:.+~ arj/ir ranraired nsreln. 8,: :ata are the, author's and NC? if::ljatior. ;. + t;lc tson or infonnatioz interchangen res~onslbiiitv of :he user t3 vaiicate and ver1.y. s integrity pr:x :3 zse.

2 DESIGN CRITERIA FOR SHAFTS AND OPEN GEARING ON MOVABLE BRIDGES Lance V. Borden, Donald L. Miller Modjeski and Masters Harrisburg, Pennsylvania INTRODUCTION Whether you are rehabilitating an old movable bridge, or designing a new replacement movable bridge, correct design procedures are necessary to insure the long life of shafts, trunnions and open gearing. AREA and AASHTO are typically noted for being conservative with their design equations and allowable stresses, but there are situations involving shafts, trunnions and open gearing where the resulting design could be flawed because fatigue, wear and stress concentrations were not adequately considered. AREA/AASHTO equations for shafts and gearing are generally based on static stress conditions. There is no or little allowance for fatigue, wear, and stress concentrations in the design equations of these mechanical components. Even with 1 i beral factors of safety based on yield strength, a static analysis/design will, in some cases, produce an improper design. The design of shafts and trunnions will be treated separately from the design of open gearing, the latter using AGMA recommended equations. SHAFT/TRUNNION DESIGN Most shafts rotate, and when they rotate the bending stresses fluctuate or reverse, producing the probability of a fatigue failure. A fatigue failure, unlike a ductile static yielding failure, occurs suddenly and usually without prior warning. The failure resembles a brittle fracture, and occurs as the result of a circumferencial crack which propagates radially inward until the remaining material can no 1 onger support the loading, hence a sudden unexpected fatiyue failure. AREA/AASHTO equation~'~.~'*for shafts and trunnions are based on static stresses only, with a K factor intended to account for rotation of the shaft. For instance: Bending Stress: (AREA 6.4.6) *See reference at end of paper.

3 Shear Stress: (AREA 6.4.6) e where M = bending moment (lb-in), T = torque (lb-in), d = shaft diameter (inch) For trunnions and counterweight shafts, K = 1. For Rotating Shafts: K & (AREA 6.4.2) where n = shaft speed in RPM. Allowable stresses given by AREA provide for stress concentrations of 140% of computed stress. A 1.4 factor is relatively small and also, no reference is made as to how to determine the stress concentration factors, or when and how to use them in the equations. The reason for ignoring stress concentrations is that with static stresses, they do not effect failure. However, since most shafts are subjected to fluctuating stresses, fatigue failure is possible, and stress concentrations have a definite effect on fatigue life. DESIGNING TO PREVENT FATIGUE FAILURE OF SHAFTS AND TRUNNIONS The important quantityto consider in fatigue design is the endurance limit of the part (S,). Steel has an endurance 1 imit, aluminum does not. Knowing S, the part can be designed so that it w i l l have millions of cycles of stress without fatigue failure. The endurance limit depends on many factors, the most important of these being: ultimate tensile strength, size, surface finish, re1 iabil ity and temperature. By eq~ation'~' (for steel subject to bending and torsion): se = 0.5 x sut x CD x cs x cr x ct x cm, S, = ultimate tensile strength (psi) C, = size (diameter) factor; for 1/2" or greater diameter, C, = 0.872(d)-',~'~~ C,= surface finish factor; for a typically machined surface (8-16 pin.): (for 60,000 5, S, 5 200,000 psi)

4 CR = reliability factor based on an 8% standard deviation on limit. Use this, especially when the ultimate tensile stress. 5. is a 'typical1' value. R = reliability. for: R = 50%, CR = 1.0 R = 90%, CR = 0.90 R = 99%, CR= 0.81 CT = temperature factor Cr = 1 for steel up to 400' F CT = 0.7 for steel at 1,000" F C, = any mi scell aneous factor eg: welding, plating, shot peening, corrosion, etc. Some design references include stress concentration fatigue factors with the endurance limit equation. Since most applications in design consist of combined bending and torsional stresses, and the stress concentration factors are different for each, it is better to include these factors with the respective. stresses (or bending and torsional Stress concentration factors depend on the shaft configuration and on the type of loading. The most common stress concentration occurs at filleted shoulders on shafts. However, other common factors are for keyways, threaded shaft portions, laterally drilled holes, or circumferencial grooves. A foremost reference of theoretical stress concentration factors (Kt for bending, K,, for torsion) is by R. E. ~eterson'". Some figures for bending and torsion of filleted shafts are included with this paper (Figures 1 and 2). The Kt and Kt, values depend on the ratio of diameters (D/d) and the ratio of fillet size to smaller diameter (r/d). These theoretical factors are modified for fatigue by a notch sensitivity factor q or q, to give Kf for bending and K,, for torsion. Kf= 1+ q Kt - 1) KL~ = (Kt, - I) are: For fillet radii greater than 1/8-inch (r > 0.12") for, S, = 80 ksi: q = 0.85, q, = 0.87 for, S, = 140 ksi: q = 0.90, q, = 0.92 approximate q values When in doubt, using q or q, Then, Kf = K, and K,, = Kt,. = 1 is conservative. The fatigue stress concentration factors and the endurance limit are then used in a design equation to solve for required shaft size. 3

5 The factor of safety (n) typically used for fatigue design is n r (Note: n in this equation is safety factor, not shaft speed RPM.) (Extracted from: FIGURE 1 "Stress Concentration Factors" by R. E. ~eterson)'~' 4

6 FIGURE 2 (Extracted from: "Stress Concentration Factors" by R. E. ~eterson)'~)

7 The above design equation is based on the Soderberg diagram fatigue failure curve and the maximum shear stress theory for combined reversing bending and 0 steady torsion stresses. For cases of non-reversing bending stresses (trunnions which rotate s 90') this equation is conservative. EXAMPLE DESIGN CALCULATIONS The following example wi 11 demonstrate the comparison between AREA/AASHTO static equations and the design based on fatigue equations. FIGURE 3 The above figure shows the shaft used in the following calculations. The example comes from the analysis of an existing design of a counterweight sheave shaft, using ASTM A668, Class D steel. The load P was e6qual to 725 kips and the torque (due to starting friction) was equal t0~1.3 x 10 lb-in. At the fillet, M=725x 10 lbx 14" = 10.15~ 1o61b-in.; T ~ 1o6lb-in.

8 Using the AREA/AASHTO equations and K = 1: a - 14,550psi < 15,000psi allowable; ok r = 7,300 psi < 7,500 psi allowable; & The conclusion would be the design is &, based on AREA/AASHTO equations. Using the previous fatigue design equation and first solving for the endurance limit., S, = 75,000 psi; S = psi (ASTM, A668, Class D) = C, = x ) C, = 1 (S, is a minimum strength and, therefore, do not adjust for reliability) S,= 0.5 x 75,000 x x x 1 S, = 18,300 psi The stress concentration factors for bending and torsion are found from the appropriate figure knowing first r/d and D/d. For Bending, Figure 1, K, = 2.6 For Torsion, Figure 2, K, = 2.0 Modifying for Fatigue (q = 0.85, q, = 0.87): Kf = 1~0.85 (2.6-1) = 2.36 Since we are analyzing a present design, rearrange the fatigue design eouation to solve for n, the factor of safety based on "indefinite" life.

9 Redesigning for an n = 1.25, the required shaft size is d = 25.6"! If we were doing a redesign, the first thing to change is the fillet radius, making it much larger so that K, and K,, would be less. Using a one-inch fillet radius, Kt = 2.0, Kt, = 1.57, Kf = 1.85, K,, = Reauired diameter is: As you can see, the original design was inadequate for long fatigue t ife, a1 though AREA/AASHTO equations say the design is satisfactory. An estimate of the expected fatigue life of the original shaft was approximately 35,000 cycles of stress. Depending on the frequency of bridge opening, this present design could have prematurely failed by fatigue. [For 20 years of life, no more than five openings per day could be allowed!] In this appl ication, the final counterweight sheave shaft will be 25 inches small diameter, 26.5 inches larger diameter and a one-inch fillet radius at the shoulder. Part of the reason for the larger shaft was the change to roller bearings from the original plain journal bronze bearings. A couple more points should be made here with regard to the design to prevent fatigue failure. 1) Even if there was no filleted shoulder where the shaft gets pressed into the counterweight sheave hub, there will still be a stress concentrati.on in the shaft at the edge of the hub. For a hub length approximately equal to the shaft diameter, the K, factor for bending can approach a value of &, when the pressure due to the pre~?~,fit is about equal to the nominal bending stress in the shaft outer surface. 2) Older bridge designs made extensive use of plain journal bronze sleeve bearings. In non-self a1 igning plain sleeve bearings, the force distribution along the length is not uniform and tends to be concentrated nearer the inner edge of the bearing, thereby reducing

10 the shaft bending moment and reducing the chance of fatigue failure. However, on many rehabilitations, the plain bearings are being replaced with self-a1 igning roller bearings. This type of bearing will cause the load to be concentrated at a larger moment arm and may, therefore, increase the chance of fatigue failure. Always check the design of the shaft carefully to be sure you are not worsening an a1 ready underdesigned situation. OPEN-GEARING DESIGN [The following will be limited to the design of open spur gearing, but procedures are similar for other types (Helical, bevel).] AREA/AASHTO for gear tooth strength (as in shaft design), use equations which are based primarily on static bending stresses of the teeth. The equations are well founded, being based on the original Lewis equation (1892), but they have long since lost their usefulness, since AGMA equations take so many more factors into account. In the latest AASHTO Standard Specification for Movable Highway Bridges (1988)'~', a sentence was added to Specification (Strength of Gear Teeth) : "Gear tooth design shall meet AGMA standards for surface durability (pitting resistance) and bending strength." The specification offers no further design guidance. It is, therefore, the purpose of this paper to compare AREA/AASHTO equations to AGMA and briefly review the appropriate AGMA standard to the reader for open spur gearing design. The AREA/AASHTO ) is: equation for 20" full -depth involute spur gears (AREA where W = allowable tooth load (1 bs.) f = effective face width (in.) s = allowable unit stress (psi) p = circular pitch (in.) n = number of teeth v = pitch line velocity (fpm) The term in the first parentheses is the Lewis form factor and is given the symbol y by many references. The term in the second parentheses is the Barth velocity factor (also originated in the nineteenth century). This factor was based on tests of cast iron gears with cast teeth, probably with cycloidal profile (not involute). The Lewis equation forms the basis for the AGMA equations in bending strength and fatigue. AGMA Standard 908"' gives the equations for both gear tooth bending strength (fatigue) and for surface durabil i ty (pitting and wear).

11 Each is treated separately, and then the final gear design is based on the more critical of the two procedures. Bending Strength (Fatigue) : Allowable Bending Stress Number: These can be combined into an AGMA power rating equation: From the previous equations: St = bending stress number (psi) W, = transmitted tangential tooth load (1 bs) Pd = diametral Pitch (l/in) F = net face width of narrowest gear (in) J = geometry factor for bending Sat = allowable bending stress number (psi) n, = pinion speed (RPM) d = pitch diameter of pinion (in.) K, = application factor for bending K, = life factor for bending K, = load distribution factor for bending K, = reliability factor for bending K, = size factor for bending KT = temperature factor for bending K, = velocity factor for bending The following equations or values for the above factors are appropriate for open gearing in movable bridges. Refer to AGMA Standard 908 for more complete figures and tab1 es. K,= K,= 1.25 (u~iform power source, moderate shock load) 1.0 (10 cycles of stress) ic,= x F x F/d x F' gearing, adjusted at assembly) K, = 1.0 (for 99% reliability) K, = 1.l (for large teeth, Pd < 6) KT = 1.0 (for gear temperatures < 250" F) (based on open

12 [ 60 ) (gear qua1 i 60 + J; ty, Q, = 6) where v = pitch line velocity of gear (ft/min.) 24iN) J = x e'--. (for tip loadings) or (-7.77/N) J = x e (for highest single tooth contact) where N = number of teeth Tip loading gives a more conservative design and is recommended for open gearing with low gear qual ity (Q, = 6). S,, = -274 t 167 x Hg x H,~ for the range H, = (for steel gears; H, = Brine11 Hardness of teeth) (For idler gears, use an S,, equal to 70% of this.) Some other equations and relations are: v = rd np/12 (ft/min) where d = pitch diameter of pinion (in) n, = speed of pinion (RPM) p = circular pitch of teeth (inch) N = number of teeth on pinion/gear d or D = pitch diameter of pinion or gear (inch) F 5 3p (this is an AREA/AASHTO limit) (AGMA gives no relation, but other references (3,5) use a range: which would be approximately 2. 5 ~ 5 ~ F 5p, Surface Durability (Pitting, Wear): Allowable Contact Stress Number:

13 These can be combined into an AGMA pitting resistance power rating equation: Where from the previous equations: S, = contact stress number (psi) Sac= allowable contact stress number (psi) W,, d, F, n, as before C, = elastic coefficient 6 C. = application factor for pitting C,= surface condition factor Cg = hardness ratio factor CL = life factor for pitting C, = load distribution factor for pitting CR = re1 iability factor for pitting C. = size factor for pitting CT = temperature factor for pitting C, = dynamic velocity factor for pitting I = geometry factor for pitting The following equations and values are appropriate for open gearing in movable bridges. Refer to AGMA 908 for more complete information. C, (AGMA gives no specific values) C, (Use greater than unity when there is poor tooth surface conditions) c, = IQ CR = 1.0 for 99% reliability C. = 1.0 (no values given by AGMA) C, = 1.0 (for gear temperatures < 250" F) C., = K. H, (Pinion) Cg = 1.0 if ~7.2 for a high hardness ratio, C, > 1 H. (Gem) C, = 1.0 FOR lii stress cycles C, = 2,300 for steel pinion steel gear C, = 2,000 for steel pinion - cast iron gear - MG I ;(for% teeth) (8.65 x MG ) For M, > 10, use the I value for M, = 10. Sac = 26,000 t 327 x HB (psi) in the range HB 5 400

14 Yieldins Criteria AGMA also presents equations for cases of infrequent momentary overload (less than 100 stress cycles). For this situation, allowable yield strength properties are the determining criteria, rather than the fatigue strength of the gear material. Wm,Ka Pd Ks Km say K, 2 x-x- K, F JKf W,,, = maximum peak tangenti a1 tooth load (I bs) K, = yield strength factor use K,. = 0.75 (Industrial Practice) or K,= 0.50 (more conservative) S,= 482 x HE - 32,800 (psi) in the range HE Kf = stress correction factor for teeth pinion, K,= 1.3 Sample Calculations To demonstrate the disparity in using the AREA/AASHTO equation for spur gear designjanalysis, rather than the AGMA equations, the following example is presented: In a particular vertical lift bridge rehabilitation project, the open hoist gearing was showing signs of excessive wear, after a relatively short life (about 20 years). An analysis revealed that the gearing design using the AREA/AASHTO equation allowed for a much hiqher horsepower ratinq - than usinq - appropriate... AGMA eauations for fatigue and wear: The input pinion driving the hoisting rope drums had the following specifications: 16 teeth, PC = 2", d = ", F = 5.5", 20" full-depth involute teeth, Material: ASTM A235, Class C1 (new specification: A668, Class C) forged steel. RPM = 28.4 meshing with a 44-tooth cast steel gear. Using the AREA/AASHTO equation and allowable stress, S (allowable) = 20,000 psi (AREA ) W = 18,940 lbs. (allowable tooth load)

15 Allowable horsepower would then be: hp = 43.5 (AREA/AASHTO Power Rating) For this same pinion, the AGMA power rating equation for bending fatigue gives the following result. For A668, Class C, an average Brine11 Hardness is H, = 160. Therefore, S,, = x x 1602 = 22,550 psi The other factors are: 60 + J; , for v - 76fpm Pa, = 24.6 hp (based on bending fatigue) This value is approximately 56% of the AREA/AASHTO a1 lowabl e horsepower. The AGMA power rating equation based on pitting resistance (surface durability and wear gives the following result:

16 S,, = 26,000 x 327 x 160 = 78,300 psi Cp = 2,300 &G C, = K, ; C, = K, = C,= Cf= Cg= Cs= CB= CL= CT= 1.0 This is about 22% of the AREA/AASHTO allowable horsepower The AGMA equation for overload yielding gives the following results: Say= 482 x ,800 = 44,300 psi K, = 0.75; K, = 1.3 Solving from W,: Using the previous K values: lbs. This value compares more closely with the AREA/AASHTO value of W = 18,940 However, AGMA says that this high a force should only be infrequent. The previous calculations should show that the AGMA equations give a much more conservative allowable horsepower rating, both for fatigue and surface durability. The original gearing on this job was designed using AREA equations and a 1.5 overload factor, the pinion tooth load being based on 30 hp. The input shaft to the pinion, however, transmitted 60 hp, since the pinion drove two gears (one through an idler pinion). Therefore, the pinion teeth were loaded twice each revolution contributing to premature wear. Also, AREA equations do not treat an idler a different, even though the teeth are subjected to reversed bending stresses every rev01 ution. For the rehabilitation, the gearing on the hoisting rope drum drive were redesigned using AGMA equations for wear and fatigue strength. Two double pinion shafts were used so that there was no idler and no pinion which was loaded twice per revolution. Material strength and face widths were increased in order to retain original circular pitches and center distances.

17 SUMMARY This paper has presented recommended design equations to be used when designing rotating shafts, trunnions, and open spur gearing for movable bridges. Using on1 y AREA/AASHTO equations, which account for static stresses alone, may lead to premature fatigue failure for shafts and premature wear or fatigue failure for spur gearing. Critical bridge machinery parts, such as these, must be careful 1 y designed, using appropriate fatigue design equations, endurance 1 imi ts, stress concentration factors, and AGMA gear equations and geometry factors. REFERENCES 1. AREA (American Railway Engineering Association) Standard, Part 6, Movable Bridges, 1955 (Reapproved with revisions 1985) 2. AASHTO (American Association of State Highway and Transportation Officials, Inc.) Standard Specifications for Movable Highway Bridges, Mechanical Engineering Design, Shigley & Mischke, 5th Ed, McGraw- Hill, I Stress Concentration Factors, R. E. Peterson, John Wiley Co, Machine Elements in Mechanical Design, R. L. Mott, C. E. Merrill Co, Handbook of Stress and Strength, Lipson & Juvinall, Macmillan, AGMA (American Gear Manufacturers Association) Standard 908, "Geometry Factors for Rating the Pitting Resistance and Bending Strength of Spur and Helical Involute Gear Teeth", 1989

12/25/ :27 PM. Chapter 14. Spur and Helical Gears. Mohammad Suliman Abuhaiba, Ph.D., PE

12/25/ :27 PM. Chapter 14. Spur and Helical Gears. Mohammad Suliman Abuhaiba, Ph.D., PE Chapter 14 Spur and Helical Gears 1 2 The Lewis Bending Equation Equation to estimate bending stress in gear teeth in which tooth form entered into the formulation: 3 The Lewis Bending Equation Assume